Analysis of convection diffusion problems at various peclet numbers using finite volume and finite difference schemes

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
16
Analysis of Convection-Diffusion Problems at Various Peclet
Numbers Using Finite Volume and Finite Difference Schemes
Anand Shukla
Department of Mathematics
Motilal Nehru National Institute of Technology, Allahabad, 211 004, U.P., India
E-mail: anandshukla86@gmail.com
Surabhi Tiwari (Corresponding author)
E-mail: surabhi@mnnit.ac.in; au.surabhi@gmail.com
Pitam Singh
E-mail: psingh11@rediffmail.com
Abstract
Convection-diffusion problems arise frequently in many areas of applied sciences and engineering. In this paper,
we solve a convection-diffusion problem by central differencing scheme, upwinding differencing scheme (which
are special cases of finite volume scheme) and finite difference scheme at various Peclet numbers. It is observed
that when central differencing scheme is applied, the solution changes rapidly at high Peclet number because
when velocity is large, the flow term becomes large, and the convection term dominates. Similarly, when
velocity is low, the diffusion term dominates and the solution diverges, i.e., mathematically the system does not
satisfy the criteria of consistency. On applying upwinding differencing scheme, we conclude that the criteria of
consistency is satisfied because in this scheme the flow direction is also considered. To support our study, a test
example is taken and comparison of the numerical solutions with the analytical solutions is done.
Keywords: Finite volume method, Partial differential equation
1. Introduction
Mathematical models of physical [2, 7], chemical, biological and environmental phenomena are governed by
various forms of differential equations. The partial differential equations describing the transport phenomena in
fluid dynamics are difficult to solve, particularly, due to the convection terms. Such equations represent the
hyperbolic conservation law for which their solutions always contain discontinuity and high gradient. Thus
accurate numerical solutions are very difficult to obtain. Special treatment must be applied to suppress spurious
oscillations of the computed solutions for both the convection and convection-dominated problems. In the
present scenario, better ways to approximate the convection term are still needed, and thus development of
accurate numerical modeling for the convection-diffusion equations remains a challenging task in computational
fluid dynamics. In recent years (see [7]), with the rapid development of energy resources and environmental
science, it is very important to study the numerical computation of underground fluid flow and the history of its
changes under heat. In actual numerical simulation, the nonlinear three-dimensional convection-dominated
diffusion problems need to be considered.
When velocity is higher, that is, flow term is larger, a simple convection-diffusion problem is converted to
convection-dominated diffusion problem because Peclet number is greater than two. A Peclet number is a
dimensionless number relevant in the study of transport phenomena in fluid flows. It is defined to be the ratio of
the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an
appropriate gradient. In the context of the transport of heat, Peclet number is equivalent to the product of
Reynolds number and Prandtl number. In the context of species or mass dispersion, Peclet number is the product
of Reynolds number and Schmidt number. For diffusion of heat (thermal diffusion), Peclet number is defined as
,e
F
P
D
=
where F is the flow term and D is the diffusion term.
In most of the problems with engineering applications (see [4]), Peclet number is often very large. In such
situations, the dependency of the flow upon downstream locations is diminished and variables in the flow tend to
become 'one-way' properties. Thus, when modeling certain situations with high Peclet numbers, simpler
computational models can be adopted. A flow can often have different Peclet numbers for heat and mass. This

17
can lead to the phenomenon of double diffusive convection. In the context of particulate motion, Peclet numbers
are also called Brenner numbers, with symbol rB , in honors of Howard Brenner.
2. Discretization Algorithms Using Central Differencing Scheme and Upwinding Differencing Scheme
The general transport equation for any fluid property is given by
( )
( ) ( ) ,div u div grad S
t
ρϕ
ρϕ ϕ ϕ
∂
+ = Γ +
∂
(1)
where ϕ is a general property of fluid, ρ is the density and u is the velocity of the fluid. In equation (1), the rate
of change term and the convective term are on the left hand side and the diffusive term (Γ is the diffusion
coefficient) and the source term S respectively are on the right hand side. In problems, where fluid flow plays a
significant role [1], we must account for the effects of convection. In nature, diffusion always occurs alongside
convection, so here we examine a method to predict combined convection and diffusion. The steady convection-
diffusion equation can be derived from transport equation (1) for a general property by deleting transient term:
( ) ( )div u div grad sρ ΦΦ = Γ Φ + (2)
Formal integration over control volume gives
.( ) ( ) .
A A CV
n u dA n grad dA S dVρ ΦΦ = Γ Φ +∫ ∫ ∫
WPxδ PExδ
wu eu
-------------------------------------------
W ------- E
P
wex δ=∆
Figure 1: P is a general point around which discussions takes place, W and E are west and east nodal points
respectively
In the absence of sources (see [5]), the steady convection and diffusion of a property in a given one dimensional
flow field u is governed by:
( ) ,
d d d
u
dx dx dx
r
骣 F ÷çF = G ÷ç ÷ç桫
(3)
and continuity equation becomes
( )
0.
d u
dx
ρ
=
Integrating equation (3), we have
( ) ( ) ,e w
e w
uA uA A A
x x
ρ ρ
∂Φ ∂Φ   
Φ − Φ = Γ − Γ   
∂ ∂    (4)
( ) ( ) 0 .e wu A u Aρ ρ− =
To obtain discretized equation, we shall take some assumptions:
Let uF ρ= represent convective mass flux per unit area and
x
D
δ
Γ
= represent diffusion conductance.
Now we are taking AAA ew == and applying the central differencing, the integrated convection-diffusion
becomes. Here u represents velocity of fluid.
( ) ( ),e e w w e E P w P WF F D DΦ − Φ = Φ −Φ − Φ −Φ (5)

18
0.e wF F− = (6)
Central Differencing Scheme
The central differencing approximation has been used to discretize the diffusion which appears in equation (5).
So ( ) ( )/ 2; / 2.e P E w W PΦ = Φ + Φ Φ = Φ + Φ
Putting these values in equation (5), we get
)()()(
2
)(
2
WPwPEePW
e
EP
e
DD
FF
Φ−Φ−Φ−Φ=Φ+Φ−Φ+Φ
E
e
eW
w
wPwe
e
e
w
w
E
e
eW
w
wP
e
e
w
w
F
D
F
DFF
F
D
F
D
F
D
F
D
F
D
F
D
Φ





−+Φ





+=Φ





−+





−+





+
Φ





−+Φ





+=Φ











−+





−
22
)(
22
2222
(7)
Equation (7) can be written as
,P P W W E Ea a aF = F + F
(8)
where Pa , Wa and Ea are coefficients of PΦ , WΦ and EΦ respectively.
3. Test Example
Example 1: The general property of a fluid Φis transported by means of convection and diffusion through the
one dimensional domain. The governing equation is given below. The boundary conditions are 10 =Φ at x=0
and 0=ΦL at x=L. Using five equally spaced cells and the central differencing scheme for convection-
diffusion problems, we calculate the distribution of Φas a function of x for the following cases:
Case (1): u=0.1 m/s
Case (2): u=2.5 m/s.
The following data are applying to obtain the solution of above problem by central differencing scheme.
Length L=1.0 m,
A xd
B
1F= 0F =
1 2 3 4 5
X=0 W w P e E X=L
xd xd
Figure 2: Discretization using five nodal points for Example (1)
Now we solve the problem by central differencing scheme and upwind differencing scheme for both the cases.
Then we compare these results by finite difference method and observe the effect of schemes on the convergence
rate of solution.
Solution of Example (1) by Central Differencing Scheme
Node Distance Finite volume
solution
Analytical
solution
Difference Percentage error
1 0.1 0.9421 0.9387 -0.003 -0.36
2 0.2 0.8006 0.7963 -0.004 -0.53
3 0.3 0.6276 0.6224 -0.005 -0.83
4 0.4 0.4163 0.4100 -0.006 -1.53
5 0.5 0.1579 0.1505 -0.007 -4.91
3
1.0 / , 0.1 / / ,kg m kg m sρ = Γ =

19
Table 1: The solution of problem by central differencing scheme for u=0.1
Table 2: The solution of problem by central differencing scheme for u=2.5
Solution of Problem by Upwind Differencing Scheme
One of the major inadequacies of the central differencing scheme is its inability to identify the flow direction.
The value of property Φ at west cell face is always influenced by both PΦ and WΦ in central differencing. In a
strongly convective flow from west to east, the above treatment is not suitable because the west cell face should
receive much stronger influencing from node W than from node P. The upwind differencing and donor cell
differencing schemes takes into account the flow direction when determining the value at cell face: the convicted
value of Φ at a cell face is taken to be equal to the value at upstream node. We show the nodal values used to
calculate cell face values when the flow is in the positive direction in Figure 2 those for the negative direction.
Figure 3: Discretization technique for upwinding differencing scheme
When the flow is in positive direction, 0, 0( 0, 0),w e w eu u F F> > > > the upwind schemes sets
.w W e PandΦ = Φ Φ = Φ ( )8
And the discretized equation (5) becomes
( ) ( ).e P w W e E P w P WF F D DΦ − Φ = Φ −Φ − Φ −Φ
This may be written as
( ) ( ) ,w e e P w w W e ED D F D F D+ + Φ = + Φ + Φ
which gives
( ) ( )[ ( )]w w e e w P w w W e ED F D F F D F D+ + + − Φ = + Φ + Φ
(9)
When flow is in the negative direction, uw<0, ue<0 (Fw <0, Fe<0), w P e EandΦ = Φ Φ = Φ
Solution
Analytical
solution
Difference Percentage error
1 0.1 1.0356 1.0000 -0.0035 -3.56
2 0.2 0.8694 0.9999 0.131 13.03
3 0.3 1.2573 0.9999 -0.257 -25.74
4 0.4 0.3521 0.9994 0.647 64.70
5 0.5 2.4644 0.9179 -1.546 -168.48

20
Figure 4: Discretization Technique for Upwinding Differencing Scheme for Example (1)
Now the discretized equation is
( ) ( )e E w P e E P w P WF F D DΦ − Φ = Φ −Φ − Φ −Φ
Or ( )[ ( ) ( )]w e e e w P w W e e eD D F F F D D F+ − + − Φ = Φ + − Φ
(10)
Identifying coefficients of WΦ
and EΦ
as aw and aE, the equation (9) and (10) can be written in the general
form as
,P P W W E Ea a aΦ = Φ + Φ (11)
with central coefficients
( ).P W E e wa a a F F= + + −
The grid shown in above figure is used for discretization. The discretization equation at internal nodes 2,3 and 4
and relevant neighbor coefficient are given by (11). Note that in this example
e w e wF F F u and D D D
x
ρ
δ
Γ
= = = = = = everywhere.
At the boundary node (1), the use of upwind differencing scheme for the convective terms gives
( ) ( ),e P A A e E P A P AF F D DΦ − Φ = Φ − Φ − Φ − Φ
and at node (5)
( ) ( ).B P w W B B P w P WF F D DΦ − Φ = Φ − Φ − Φ − Φ
At the boundary nodes we have
2
2A B A BD D D and F F F
xδ
Γ
= = = = = and as usual boundary condition
enter the discretized equation as source contribution:
P P W W E E ua a a SΦ = Φ + Φ +
with
( ) .P W E e w Pa a a F F S= + + − −
Node aW aE SP Su
1 0 D -(2D+F) (2 ) AD F+ Φ
2,3,4 D+F D 0 0
5 D+F 0 -2D 2 BDΦ
Table 3: Determination of coefficients aw, aE, Sp and Su

21
Case 1: U=0.1 m/s: 0.1, 0.1/ 0.2 0.5 0.2e
F
F u D so P
x D
ρ
δ
Γ
= = = = = = = where Pe
is known as Peclet
number. Now coefficients are given in the table as follow:
Node aW aE SP Su aP
1 0 0.5 -1.1 1.1 1.6
2,3,4 0.6 0.5 0 0 1.6
5 0.6 0 -1 0 1.6
Table 4: Determination of coefficients aw, aE, Sp, Su and aP for U=0.1 m/s
1 2
2 1 3
3 2 4
4 3 5
5 4
1.6 0.5 1.1
1.1 0.6 0.5
1.1 0.6 0.5
1.1 0.6 0.5
1.6 0.6 1.6
Φ = Φ +
Φ = Φ + Φ
Φ = Φ + Φ
Φ = Φ + Φ
Φ = Φ +
1.6000牋 0.5000牋牋牋牋0 牋牋牋牋0牋牋牋牋 0
? .6000牋? .1000牋0.5000牋牋牋牋0牋牋牋? ? 牋?
?牋 0.6000牋? .1000牋 0.5000牋牋牋牋0
?牋牋牋牋 0牋 0.6000牋? .1000牋 0.5000牋?
牋?牋牋牋牋0牋 0 0.6000牋? .6000
−
− −
− −
− −
−
1
2
3
4
5
1?1000牋
0牋?
0牋牋?
0牋牋牋
0
Φ    
    Φ    
    Φ =
    
Φ    
    Φ    
Thus
1
2
3
4
5
0.9337牋?
0.7879牋?
0.6130?
0.4031牋?
0.1512
Φ   
   Φ   
   Φ =
   
Φ   
   Φ   
Node Distance Upwind solution Analytical
solution
Central
Differencing
solution
Finite
difference
solution
1 0.1 0.9337牋? 0.9387 0.9421 0.9048
2 0.2 0.7879牋? 0.7963 0.8006 0.7884
3 0.3 0.6130? 0.6224 0.6276 0.6461
4 0.4 0.4031牋? 0.4100 0.4163 0.4722
5 0.5 0.1512 0.1505 0.1579 0.2597
Table 4: Comparison for various results for U=0.1 m/s
Case 2: U=2.5 m/s: 2.5, 0.1/ 0.2 0.5 5e
F
F u D so P
x D
ρ
δ
Γ
= = = = = = = where Pe >2. Due to this
reason upwinding differencing scheme provide better convergence. In this case, the coefficients are given below:

22
Node aW aE SP Su aP
1 0 0.5 -3.5 3.5 4.0
2,3,4 3.0 0.5 0 0 3.5
5 3.0 0 -1.0 0 4.0
Table 7: Determination of coefficients aw, aE, Sp, Su and aP for U=2.5 m/s
1 2
2 1 3
3 2 4
4 3 5
5 4
4 0.5 3.5
3.5 3 0.5
3.5 3 0.5
3.5 3 0.5
4 3
Φ = Φ +
Φ = Φ + Φ
Φ = Φ + Φ
Φ = Φ + Φ
Φ = Φ
4.0000 -0.5000 0 0 0
-3.0000 3.5000 -0.5000 0 0
0 -3.0000 3.5000 -0.5000 0
0 0 -3.0000 3.5000 -0.5000
0 0 0 -3.0000
1
2
3
4
5
?.5000牋
0牋?
0牋牋?
0牋牋牋
4.0000 0
Φ    
    Φ    
    Φ =
    
Φ    
    Φ    
1
2
3
4
5
0.9998
0.9987
0.9921
0.9524
0.7143
Φ   
   Φ   
   Φ =
   
Φ   
   Φ   
4. Discretization Algorithm for Finite Difference Method
For solving any given boundary value problems, we can divide the range 0, nx x轾
臌 in to n equal subintervals
of width h so that 0 ,ix x ih= + i=1, 2… n.
The finite difference approximations of derivatives of any fluid property F at ix x= are given by
1 1 2
' ( )
2
i i
i O h
h
+ -F - F
F = + And 21 1
2
2
'' ( )i i i
i o h
h
- -F - F + F
F = +
On the basis of above discussion, we now solve the given convection dominated diffusion problem in Example
(1) by finite difference method for equal spacing.
Node Distance Upwinding solution Central Differencing
Solution
Analytical solution Finite difference solution
1 0.1 0.9998 1.0356 1.0000 1.0208
2 0.2 0.9987
0.8694
0.9999 0.9723
3 0.3 0.9921
1.2573
0.9999 1.0854
4 0.4 0.9524 0.3521 0.9994 0.8214
5 0.5 0.7143 2.4644 0.9179 1.4375

23
Case1: U=0.1
Solution
Analytical solution Finite difference
solution
1 0.1 0.9421 0.9387 0.9048
2 0.2 0.8006 0.7963 0.7884
3 0.3 0.6276 0.6224 0.6461
4 0.4 0.4163 0.4100 0.4722
5 0.5 0.1579 0.1505 0.2597
Case2: U=2.5
5. Graphical Representation of Convergence for Example (1)
Case 1: U=0.1
The graphical result of Table 9 and Table 10 are shown in fig. 5 which describes convergence criteria for various
methods
Figure 5: Convergence criteria for various methods for U=0.1
Solution
Analytical solution Finite difference
solution
1 0.1 1.0356 1.0000 1.0208
2 0.2 0.8694 0.9999 0.9723
3 0.3 1.2573 0.9999 1.0854
4 0.4 0.3521 0.9994 0.8214
5 0.5 2.4644 0.9179 1.4375

24
Case2: U=2.5
Figure 6: Convergence criteria for various methods for U=2.5
6. Concluding Remark
When U=2.5, we observe that result show better convergence since
2
e
E e
F
a D= − and the convective
contribution to the east coefficient is negative. Thus, the solution converges faster towards the exact solution for
large Peclet number. If the convection dominates, aE can be negative. Given that Fw >0 and Fe >0 (i.e., flow is
unidirectional), for aE to be positive De and Fe must satisfy the condition: 2.e
e
e
F
p
D
= < If Pe is greater than 2
the east coefficient will be negative. This violates one of the requirements for boundedness and may lead to
physically impossible solution.
References
[1] J. D. Anderson, “Computational Fluid Dynamics: The Basics with Applications”, McgrawHil, Edition 6,
1995.
[2] M. Kumar, S Tiwari, “An initial value technique to solve third-order reaction diffusion singularly
perturbed boundary value problem”, Internat. J. Comp. Math., 89 (17) (2012), 2345-2352, doi:
10.1080/00207160.2012.706280
[3] G. Manzini, A. Russo, “A finite volume method for advection–diffusion problems in convection-
dominated regimes”, Comput. Methods Appl. Mech. Engrg. 197 (2008), 1242–1261.
[4] A. Rodrıguez-Ferran, M. L. Sandoval, “Numerical performance of incomplete factorizations for 3D
transient convection–diffusion problems”, Advances in Engineering Software 38 (2007) 439–450.
[5] N. Sanın, G. Montero, “A finite difference model for air pollution simulation”, Advances in
Engineering Software 38 (2007) 358–365.
[6] H. Versteeg, W. Malalasekra, “An Introduction to Computational Fluid Dynamics: The Finite Volume
Method”, 1995.
[7] X. Wang, “Velocity/pressure mixed finite element and finite volume formulation with, ALE
descriptions for nonlinear fluid-structure interaction problems”, Advances in Engineering Software 31
(2000) 35–44.

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Analysis of convection diffusion problems at various peclet numbers using finite volume and finite difference schemes

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Analysis of convection diffusion problems at various peclet numbers using finite volume and finite difference schemes